In this paper, and inspired by the recent discrete-time model in [1,2], we study two continuous-time opinion dynamics models (Model 1 and Model 2) where the individuals discuss opinions on multiple logically interdependent topics. The logical interdependence between the different topics is captured by a `logic' matrix, which is distinct from the Laplacian matrix capturing interactions between individuals. For each of Model 1 and Model 2, we obtain a necessary and sufficient condition for the network to reach to a consensus on each separate topic. The condition on Model 1 involves a combination of the eigenvalues of the logic matrix and Laplacian matrix, whereas the condition on Model 2 requires only separate conditions on the logic matrix and Laplacian matrix. Further investigations of Model 1 yields two sufficient conditions for consensus, and allow us to conclude that one way to guarantee a consensus is to reduce the rate of interaction between individuals exchanging opinions. By placing further restrictions on the logic matrix, we also establish a set of Laplacian matrices which guarantee consensus for Model 1. The two models are also expanded to include stubborn individuals, who remain attached to their initial opinions. Sufficient conditions are obtained for guaranteeing convergence of the opinion dynamics system, with the final opinions generally being at a persistent disagreement. Simulations are provided to illustrate the results.

中文翻译：

---

多个相互依存主题的连续时间观点动态

在本文中，受 [1,2] 中最近的离散时间模型的启发，我们研究了两个连续时间观点动态模型（模型 1 和模型 2），其中个人讨论多个逻辑上相互依赖的主题的观点。不同主题之间的逻辑相互依赖性由“逻辑”矩阵捕获，该矩阵与捕获个体之间交互的拉普拉斯矩阵不同。对于模型 1 和模型 2 中的每一个，我们都获得了网络就每个单独的主题达成共识的充分必要条件。模型 1 上的条件涉及逻辑矩阵和拉普拉斯矩阵的特征值的组合，而模型 2 上的条件只需要逻辑矩阵和拉普拉斯矩阵上的单独条件。对模型 1 的进一步研究产生了两个达成共识的充分条件，并让我们得出结论，保证达成共识的一种方法是降低个人交换意见之间的互动率。通过对逻辑矩阵进行进一步的限制，我们还建立了一组拉普拉斯矩阵，以保证模型 1 的共识。这两个模型也被扩展到包括顽固的个体，他们仍然依附于他们的初始意见。获得了足够的条件来保证意见动态系统的收敛，最终意见通常处于持续分歧。提供了模拟来说明结果。我们还建立了一组拉普拉斯矩阵来保证模型 1 的共识。这两个模型也被扩展到包括固执的个人，他们仍然依附于他们的初始意见。获得了足够的条件来保证意见动态系统的收敛，最终意见通常处于持续分歧。提供了模拟来说明结果。我们还建立了一组拉普拉斯矩阵来保证模型 1 的共识。这两个模型也被扩展到包括固执的个人，他们仍然依附于他们的初始意见。获得了足够的条件来保证意见动态系统的收敛，最终意见通常处于持续分歧。提供了模拟来说明结果。